Tunnel Through Earth: Particle Motion Facts

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SUMMARY

A particle dropped from a height equal to the radius of the Earth into a tunnel along a diameter will oscillate through the Earth to a height R on both sides, execute periodic motion, and pass through the center of the Earth with a speed of √(2GM/R). The particle's motion is periodic due to the conservation of energy, confirming statements (A), (C), and (D) as true. However, statement (B) is incorrect because the gravitational field inside the Earth varies as GMr/R^3, differing from the GM/r^2 field above the Earth's surface, thus disqualifying the motion as simple harmonic.

PREREQUISITES
  • Understanding of gravitational fields, specifically GM/r^2 and GMr/R^3.
  • Knowledge of conservation of energy principles in physics.
  • Familiarity with periodic motion and its characteristics.
  • Basic concepts of simple harmonic motion (SHM).
NEXT STEPS
  • Study the principles of gravitational fields and their variations inside celestial bodies.
  • Research the conservation of energy in oscillatory systems.
  • Explore the characteristics of periodic motion in physics.
  • Learn about the conditions for simple harmonic motion and its mathematical formulation.
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Students of physics, educators teaching mechanics, and anyone interested in gravitational dynamics and oscillatory motion.

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Homework Statement


A particle is dropped from a height equal to the radius of the Earth into a tunnel dug through the Earth along one of the diameters. Which of the following is/are true--
(A) Particle will oscillate through the Earth to a height R on both sides
(B) Particle will execute simple harmonic motion
(C) Motion of the particle is periodic
(D) Particle passes through the centre of Earth with a speed √ (2GM/R)

Homework Equations

The Attempt at a Solution


The answers are (A), (C), (D).

I know A and C will be true, A due to conservation of energy and C because such motions are almost always periodic, but I don't get why (B) is wrong and (D) is true.
Is (B) wrong because field above the Earth is GM/r^2 and inside the Earth GMr/R^3 and these are different, so it won't be SHM?
In any case, I can't prove (D) to be true (I tried energy conservation)
 
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